are produced in a forging process, and the diameter is a critical factor that must be controlled. From sample sizes of $10$ pistons produced each day, the mean and the range of this diameter have been as follows:

$\backslash$table$[[$DAY$,$MEAN $($MM$),$RANGE $($MM$)],[1,156\mathrm{.}9,4\mathrm{.}2],[2,153\mathrm{.}2,4\mathrm{.}6],[3,153\mathrm{.}6,4\mathrm{.}1],[4,155\mathrm{.}5,5\mathrm{.}0],[5,156\mathrm{.}6,4\mathrm{.}5]]$

a$)$ What is the value of $\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})?$

b$)$ What is the value of $\frac{?}{\text{b}\text{a}\text{r}}(R)?$

c$)$ What are the $UC{L}_{\stackrel{}{x}}$ and $LC{L}_{\stackrel{}{x}},$ using $3?$ Plot the data.

d$)$ What are the $UC{L}_R$ and $LC{L}_R,$ using $3?$ Plot the data.

e$)$ If the true diameter mean should be $155mm$ and you want this as your center $($nominal$)$ line, what are the new $UC{L}_{\stackrel{}{x}}$ and $LC{L}_{\stackrel{}{x}}?Px$